I strongly recommend this excellent book for an undergraduate or first-year graduate course in complex variables. One typically judges undergraduate books by a) richness of exercises, b) exposition of theory, c) the presence of supplemental topics, d) diversity of applications, e) graphing accompaniment, f) friendly aids, g) references and h) other non-standard book attributes. This book excels in all eight areas. Let us briefly review each one.

**Richness of exercises**: Perhaps the strongest point of this book is the novelty in design. *Complex Variables* combines two texts into one: There are 10 chapters presenting the theory of complex variables along with many illustrative examples. This is followed by book II, 10 chapters of problem sessions including an entire gamut of problems from routine plug-in exercises to challenging problems and projects. Book II has answers, worked out solutions, an interesting classification system of problem diversity as well as friendly aids to the struggling student (Which we discuss below).

**Exposition of Theory**: The standard bread and butter complex variable topics are all present — the complex plane, De Moivre, differentiability, Cauchy-Riemann, contour integration, residue theory, Cauchy residue theory, branch points, bilinear transformations, conformal mappings, and applications such as fluid flow.

**Supplemental Topics**: The text has an entire chapter on asymptotic expansions, a topic not normally covered in depth in complex variable texts. This chapter presents Laplace’s method, the method of steepest descent, the saddle point method and of course Mittag-Leffler. Other supplemental topics scattered throughout the book include points at infinity, stereographic projection, harmonic functions, the Schwartz reflection principle, Schwartz-Chrristoffel transformations and Joukowski maps.

**Diversity of Applications**: Many complex variable texts present applications in illustrative examples mingled with the main text. *Complex Variables *has an entire chapter devoted to applications including the standard traditional boundary value problems, heat flow, fluid flow and electrostatics. Gauss’s law and Bernoulli’s theorem are also explicitly stated and explored.

**Graphing**: Graphing is the watchword these days in calculus and linear algebra textbooks. It is therefore refreshing to find in *Complex Variables* many illustrative graphing techniques including 3-dimensional surface plots, level curves and Pólya fields. The sections on conformal mappings and bilinear transformations accompany many examples with plane-to-plane graphs.

**Friendly aids**: The problem sessions are named *Tutorial, Exercise, Question, Quiz, Open-Ended, Mixed Bag*, etc. The *Tutorial, Exercise*, and *Question* sets are primarily intended for beginners, and an experienced reader may skip them or just glance through them. The sets titled *Quiz, Questions*, and *Mined* are for those who want to test their understanding of a topic. An experienced reader looking for something new may skip these sets and go directly to *Open-Ended, Mixed-Bag* and *Applied* sessions. An important feature of these sessions is that usually each session focuses on a small number of theoretical concepts or problem-solving strategies.

**References**: Throughout the book the author cites relevant URLs particularly for graphs as well as other useful references. For example, the following URL presents 3-dimensional plots of functions of a complex variable on the Riemann sphere: http://functions.wolfram.com/ElementaryFunctions/Sech/visualizations/9/ (but note that *Complex Variables* lists *...visualizations/9.html* as the URL when in fact ...*visualizations/9/* is what works). The following URL presents a unique mathematics visualization program: http://rsp.math.brandeis.edu/3D-XplorMath/j/index.html (This URL is not directly given in the book; however it was easy to identify the URL by looking up the prefix of a broken URL in the book) The author recommends W. R. Lepage’s *Complex Variables and Laplace Transforms for Engineers* to learn more about use of Riemann surfaces in various applications:

**Other non-standard book attributes:** The author skillfully uses tables throughout the book, for example by suggesting a specific tabular form for answers in the problem book. Finally the author has a sense of humor: For example, section 6.11 presents 10 residue computations for difficult functions such as exp(1/z)/(z(z-1)z) at z = 0. The author comments, “If you can compute all the problems in 20 minutes, you deserve a medal!”

Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

## Comments

## Suvodip Mukherjee

This book gives a very neat approach to complex variables. The doubts that a student is likely to have are addressed nicely, and as a result readers' concepts will be very good. There is a huge number of exercises, which are designed in such a way that helps the reader understand the chapters properly, I can say that this book is unique in this field because of its completeness through good explanations of the theory part and a huge package of problems. Its importance can be understood by any student studying for the MSc.